On the number of generalized Sidon sets

A set A of nonnegative integers is called a Sidon set if there is no Sidon 4-tuple, i.e., (a, b, c, d) in A with a + b = c + d and {a, b} ∩ {c, d} = ∅. Cameron and Erdős proposed the problem of determining the number of Sidon sets in [n]. Results of Kohayakawa, Lee, Rödl and Samotij, and Saxton and Thomason have established that the number of Sidon sets is between 2⁽¹.16+o(1))√ⁿ and 2⁽⁶.442+o(1))√ⁿ. An α-generalized Sidon set in [n] is a set with at most α Sidon 4-tuples. One way to extend the problem of Cameron and Erdős is to estimate the number of α-generalized Sidon sets in [n]. We show that the number of (n/ log⁴ n)-generalized Sidon sets in [n] with additional restrictions is 2^Θ(√n) . In particular, the number of (n/ log⁵ n)-generalized Sidon sets in [n] is 2^Θ(√n) . Our approach is based on some variants of the graph container method.